Xenharmonic Wiki

53edo: Difference between revisions

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=Theory=
== Theory ==
The famous ''53 equal division'' divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[5-limit|5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the [[Optimal_patent_val|optimal patent val]] for [[Nuwell_family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[Semicomma_family|orwell temperament]], which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for [[Marvel_family|athene temperament]]. It is the eighth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime_numbers|prime]] edo, following [[47edo|47edo]] and coming before [[59edo|59edo]].
The famous ''53 equal division'' divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[5-limit|5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the [[Optimal_patent_val|optimal patent val]] for [[Nuwell_family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[Semicomma_family|orwell temperament]], which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for [[Marvel_family|athene temperament]]. It is the eighth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime_numbers|prime]] edo, following [[47edo]] and coming before [[59edo]].


53EDO has also found a certain dissemination as an EDO tuning for [[Arabic,_Turkish,_Persian|Arabic/Turkish/Persian music]].
53EDO has also found a certain dissemination as an EDO tuning for [[Arabic,_Turkish,_Persian|Arabic/Turkish/Persian music]].
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[http://en.wikipedia.org/wiki/53_equal_temperament Wikipedia article about 53edo]
[http://en.wikipedia.org/wiki/53_equal_temperament Wikipedia article about 53edo]


=Intervals=
== Intervals ==
{| class="wikitable"
{| class="wikitable center-all right-3 left-4 left-6"
|-
|-
! | degree
! #
! | solfege
! Solfege
! | cents
! Cents
! | approximate ratios
! Approximate ratios
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
! colspan="3" | [[Ups_and_Downs_Notation|Ups and downs notation]]
! | generator for
! Generator for
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | do
| do
| style="text-align:center;" | 0.00
| 0.00
| style="text-align:center;" | 1/1
| [[1/1]]
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | unison
| unison
| style="text-align:center;" | D
| D
| |  
|  
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | di
| di
| style="text-align:center;" | 22.64
| 22.64
| style="text-align:center;" | 81/80, 64/63, 50/49
| [[81/80]], [[64/63]], [[50/49]]
| style="text-align:center;" | ^1
| ^1
| style="text-align:center;" | up unison
| up unison
| style="text-align:center;" | ^D
| ^D
| |  
|  
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | daw
| daw
| style="text-align:center;" | 45.28
| 45.28
| style="text-align:center;" | 49/48, 36/35, 33/32, 128/125
| [[49/48]], [[36/35]], [[33/32]], [[128/125]]
| style="text-align:center;" | ^^1,
| ^^1, <br>vvm2
 
| double-up unison, <br>double-down minor 2nd
vvm2
| ^^D, <br>vvEb
| style="text-align:center;" | double-up unison,
| [[Quartonic]]
 
double-down minor 2nd
| style="text-align:center;" | ^^D,
 
vvEb
| | [[Quartonic|Quartonic]]
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | ro
| ro
| style="text-align:center;" | 67.925
| 67.92
| style="text-align:center;" | 27/26, 26/25, 25/24, 22/21
| [[27/26]], [[26/25]], [[25/24]], [[22/21]]
| style="text-align:center;" | vm2
| vm2
| style="text-align:center;" | downminor 2nd
| downminor 2nd
| style="text-align:center;" | vEb
| vEb
| |  
|  
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | rih
| rih
| style="text-align:center;" | 90.57
| 90.57
| style="text-align:center;" | 21/20, 256/243
| [[21/20]], [[256/243]]
| style="text-align:center;" | m2
| m2
| style="text-align:center;" | minor 2nd
| minor 2nd
| style="text-align:center;" | Eb
| Eb
| |  
|  
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | ra
| ra
| style="text-align:center;" | 113.21
| 113.21
| style="text-align:center;" | 16/15, 15/14
| [[16/15]], [[15/14]]
| style="text-align:center;" | ^m2
| ^m2
| style="text-align:center;" | upminor 2nd
| upminor 2nd
| style="text-align:center;" | ^Eb
| ^Eb
| |  
|  
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | 135.85
| 135.85
| style="text-align:center;" | 14/13, 13/12, 27/25
| [[14/13]], [[13/12]], [[27/25]]
| style="text-align:center;" | v~2
| v~2
| style="text-align:center;" | downmid 2nd
| downmid 2nd
| style="text-align:center;" | ^^Eb
| ^^Eb
| |  
|  
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | ruh
| ruh
| style="text-align:center;" | 158.49
| 158.49
| style="text-align:center;" | 12/11, 11/10, 800/729
| [[12/11]], [[11/10]], [[800/729]]
| style="text-align:center;" | ^~2
| ^~2
| style="text-align:center;" | upmid 2nd
| upmid 2nd
| style="text-align:center;" | vvE
| vvE
| | [[Hemikleismic|Hemikleismic]]
| [[Hemikleismic]]
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | reh
| reh
| style="text-align:center;" | 181.13
| 181.13
| style="text-align:center;" | 10/9
| [[10/9]]
| style="text-align:center;" | vM2
| vM2
| style="text-align:center;" | downmajor 2nd
| downmajor 2nd
| style="text-align:center;" | vE
| vE
| |  
|  
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | re
| re
| style="text-align:center;" | 203.77
| 203.77
| style="text-align:center;" | 9/8
| [[9/8]]
| style="text-align:center;" | M2
| M2
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | E
| E
| |  
|  
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | ri
| ri
| style="text-align:center;" | 226.42
| 226.42
| style="text-align:center;" | 8/7, 256/225
| [[8/7]], [[256/225]]
| style="text-align:center;" | ^M2
| ^M2
| style="text-align:center;" | upmajor 2nd
| upmajor 2nd
| style="text-align:center;" | ^E
| ^E
| |  
|  
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | raw
| raw
| style="text-align:center;" | 249.06
| 249.06
| style="text-align:center;" | 15/13, 144/125
| [[15/13]], [[144/125]]
| style="text-align:center;" | ^^M2,
| ^^M2, <br>vvm3
 
| double-up major 2nd, <br>double-down minor 3rd
vvm3
| ^^E, <br>vvF
| style="text-align:center;" | double-up major 2nd,
| [[Hemischis]]
 
double-down minor 3rd
| style="text-align:center;" | ^^E,
 
vvF
| | [[Hemischis|Hemischis]]
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | ma
| ma
| style="text-align:center;" | 271.70
| 271.70
| style="text-align:center;" | 7/6, 75/64
| [[7/6]], [[75/64]]
| style="text-align:center;" | vm3
| vm3
| style="text-align:center;" | downminor 3rd
| downminor 3rd
| style="text-align:center;" | vF
| vF
| | [[Orwell|Orwell]]
| [[Orwell]]
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | meh
| meh
| style="text-align:center;" | 294.34
| 294.34
| style="text-align:center;" | 13/11, 32/27
| [[13/11]], [[32/27]]
| style="text-align:center;" | m3
| m3
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | F
| F
| |  
|  
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | me
| me
| style="text-align:center;" | 316.98
| 316.98
| style="text-align:center;" | 6/5
| [[6/5]]
| style="text-align:center;" | ^m3
| ^m3
| style="text-align:center;" | upminor 3rd
| upminor 3rd
| style="text-align:center;" | ^F
| ^F
| | [[Hanson|Hanson]]/[[catakleismic|Catakleismic]]
| [[Hanson]]/[[Catakleismic]]
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | mu
| mu
| style="text-align:center;" | 339.62
| 339.62
| style="text-align:center;" | 11/9, 243/200
| [[11/9]], [[243/200]]
| style="text-align:center;" | v~3
| v~3
| style="text-align:center;" | downmid 3rd
| downmid 3rd
| style="text-align:center;" | ^^F
| ^^F
| | [[Amity|Amity]]/[[Hitchcock|Hitchcock]]
| [[Amity]]/[[Hitchcock]]
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | muh
| muh
| style="text-align:center;" | 362.26
| 362.26
| style="text-align:center;" | 16/13, 100/81
| [[16/13]], [[100/81]]
| style="text-align:center;" | ^~3
| ^~3
| style="text-align:center;" | upmid 3rd
| upmid 3rd
| style="text-align:center;" | vvF#
| vvF#
| |  
|  
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | mi
| mi
| style="text-align:center;" | 384.91
| 384.91
| style="text-align:center;" | 5/4
| [[5/4]]
| style="text-align:center;" | vM3
| vM3
| style="text-align:center;" | downmajor 3rd
| downmajor 3rd
| style="text-align:center;" | vF#
| vF#
| |  
|  
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | maa
| maa
| style="text-align:center;" | 407.55
| 407.55
| style="text-align:center;" | 81/64
| [[81/64]]
| style="text-align:center;" | M3
| M3
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | F#
| F#
| |  
|  
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | mo
| mo
| style="text-align:center;" | 430.19
| 430.19
| style="text-align:center;" | 9/7, 14/11
| [[9/7]], [[14/11]]
| style="text-align:center;" | ^M3
| ^M3
| style="text-align:center;" | upmajor 3rd
| upmajor 3rd
| style="text-align:center;" | ^F#
| ^F#
| | [[Hamity|Hamity]]
| [[Hamity]]
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | maw
| maw
| style="text-align:center;" | 452.83
| 452.83
| style="text-align:center;" | 13/10, 125/96
| [[13/10]], [[125/96]]
| style="text-align:center;" | ^^M3,
| ^^M3, <br>vv4
 
| double-up major 3rd, <br>double-down 4th
vv4
| ^^F#, <br>vvG
| style="text-align:center;" | double-up major 3rd,
|  
 
double-down 4th
| style="text-align:center;" | ^^F#,
 
vvG
| |  
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | fe
| fe
| style="text-align:center;" | 475.47
| 475.47
| style="text-align:center;" | 21/16, 675/512, 320/243
| [[21/16]], [[675/512]], [[320/243]]
| style="text-align:center;" | v4
| v4
| style="text-align:center;" | down 4th
| down 4th
| style="text-align:center;" | vG
| vG
| | [[Vulture|Vulture]]/[[Buzzard|Buzzard]]
| [[Vulture]]/[[Buzzard]]
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:center;" | fa
| fa
| style="text-align:center;" | 498.11
| 498.11
| style="text-align:center;" | 4/3
| [[4/3]]
| style="text-align:center;" | P4
| P4
| style="text-align:center;" | perfect 4th
| perfect 4th
| style="text-align:center;" | G
| G
| |  
|  
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | fih
| fih
| style="text-align:center;" | 520.75
| 520.75
| style="text-align:center;" | 27/20
| [[27/20]]
| style="text-align:center;" | ^4
| ^4
| style="text-align:center;" | up 4th
| up 4th
| style="text-align:center;" | ^G
| ^G
| |  
|  
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | fu
| fu
| style="text-align:center;" | 543.40
| 543.40
| style="text-align:center;" | 11/8, 15/11
| [[11/8]], [[15/11]]
| style="text-align:center;" | v~4
| v~4
| style="text-align:center;" | downmid 4th
| downmid 4th
| style="text-align:center;" | ^^G
| ^^G
| |  
|  
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:center;" | fuh
| fuh
| style="text-align:center;" | 566.04
| 566.04
| style="text-align:center;" | 18/13
| [[18/13]]
| style="text-align:center;" | ^~4,
| ^~4, <br>vd5
 
| upmid 4th, <br>downdim 5th
vd5
| vvG#, <br>vAb
| style="text-align:center;" | upmid 4th,
| [[Tricot]]
 
downdim 5th
| style="text-align:center;" | vvG#,
 
vAb
| | [[Tricot|Tricot]]
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:center;" | fi
| fi
| style="text-align:center;" | 588.68
| 588.68
| style="text-align:center;" | 7/5, 45/32
| [[7/5]], [[45/32]]
| style="text-align:center;" | vA4,
| vA4, <br>d5
 
| downaug 4th, <br>dim 5th
d5
| vG#, <br>Ab
| style="text-align:center;" | downaug 4th,
|  
 
dim 5th
| style="text-align:center;" | vG#,
 
Ab
| |  
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:center;" | se
| se
| style="text-align:center;" | 611.32
| 611.32
| style="text-align:center;" | 10/7, 64/45
| [[10/7]], [[64/45]]
| style="text-align:center;" | A4,
| A4, <br>^d5
 
| aug 4th, <br>updim 5th
^d5
| G#, <br>^Ab
| style="text-align:center;" | aug 4th,
|  
 
updim 5th
| style="text-align:center;" | G#,
 
^Ab
| |  
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:center;" | suh
| suh
| style="text-align:center;" | 633.96
| 633.96
| style="text-align:center;" | 13/9
| [[13/9]]
| style="text-align:center;" | ^A4,
| ^A4, <br>v~5
 
| upaug 4th, <br>downmid 5th
v~5
| ^G#, <br>^^Ab
| style="text-align:center;" | upaug 4th,
|  
 
downmid 5th
| style="text-align:center;" | ^G#,
 
^^Ab
| |  
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:center;" | su
| su
| style="text-align:center;" | 656.60
| 656.60
| style="text-align:center;" | 16/11, 22/15
| [[16/11]], [[22/15]]
| style="text-align:center;" | ^~5
| ^~5
| style="text-align:center;" | upmid 5th
| upmid 5th
| style="text-align:center;" | vvA
| vvA
| |  
|  
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:center;" | sih
| sih
| style="text-align:center;" | 679.25
| 679.25
| style="text-align:center;" | 40/27
| [[40/27]]
| style="text-align:center;" | v5
| v5
| style="text-align:center;" | down 5th
| down 5th
| style="text-align:center;" | vA
| vA
| |  
|  
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:center;" | sol
| sol
| style="text-align:center;" | 701.89
| 701.89
| style="text-align:center;" | 3/2
| [[3/2]]
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | perfect 5th
| perfect 5th
| style="text-align:center;" | A
| A
| | [[Helmholtz|Helmholtz]]/[[Garibaldi|Garibaldi]]
| [[Helmholtz]]/[[Garibaldi]]
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:center;" | si
| si
| style="text-align:center;" | 724.53
| 724.53
| style="text-align:center;" | 32/21, 243/160, 1024/675
| 32/21, 243/160, 1024/675
| style="text-align:center;" | ^5
| ^5
| style="text-align:center;" | up 5th
| up 5th
| style="text-align:center;" | ^A
| ^A
| |  
|  
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:center;" | saw
| saw
| style="text-align:center;" | 747.17
| 747.17
| style="text-align:center;" | 20/13, 192/125
| [[20/13]], [[192/125]]
| style="text-align:center;" | ^^5,
| ^^5, <br>vvm6
 
| double-up 5th, <br>double-down minor 6th
vvm6
| ^^A, <br>vvBb
| style="text-align:center;" | double-up 5th,
|  
 
double-down minor 6th
| style="text-align:center;" | ^^A,
 
vvBb
| |  
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:center;" | lo
| lo
| style="text-align:center;" | 769.81
| 769.81
| style="text-align:center;" | 14/9, 25/16, 11/7
| [[14/9]], [[25/16]], [[11/7]]
| style="text-align:center;" | vm6
| vm6
| style="text-align:center;" | downminor 6th
| downminor 6th
| style="text-align:center;" | vBb
| vBb
| |  
|  
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:center;" | leh
| leh
| style="text-align:center;" | 792.45
| 792.45
| style="text-align:center;" | 128/81
| [[128/81]]
| style="text-align:center;" | m6
| m6
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | Bb
| Bb
| |  
|  
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:center;" | le
| le
| style="text-align:center;" | 815.09
| 815.09
| style="text-align:center;" | 8/5
| [[8/5]]
| style="text-align:center;" | ^m6
| ^m6
| style="text-align:center;" | upminor 6th
| upminor 6th
| style="text-align:center;" | ^Bb
| ^Bb
| |  
|  
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:center;" | lu
| lu
| style="text-align:center;" | 837.74
| 837.74
| style="text-align:center;" | 13/8, 81/50
| [[13/8]], [[81/50]]
| style="text-align:center;" | v~6
| v~6
| style="text-align:center;" | downmid 6th
| downmid 6th
| style="text-align:center;" | ^^Bb
| ^^Bb
| |  
|  
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:center;" | luh
| luh
| style="text-align:center;" | 860.38
| 860.38
| style="text-align:center;" | 18/11, 400/243
| [[18/11]], [[400/243]]
| style="text-align:center;" | ^~6
| ^~6
| style="text-align:center;" | upmid 6th
| upmid 6th
| style="text-align:center;" | vvB
| vvB
| |  
|  
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:center;" | la
| la
| style="text-align:center;" | 883.02
| 883.02
| style="text-align:center;" | 5/3
| [[5/3]]
| style="text-align:center;" | vM6
| vM6
| style="text-align:center;" | downmajor 6th
| downmajor 6th
| style="text-align:center;" | vB
| vB
| |  
|  
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:center;" | laa
| laa
| style="text-align:center;" | 905.66
| 905.66
| style="text-align:center;" | 22/13, 27/16
| [[22/13]], [[27/16]]
| style="text-align:center;" | M6
| M6
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | B
| B
| |  
|  
|-
|-
| style="text-align:center;" | 41
| 41
| style="text-align:center;" | lo
| lo
| style="text-align:center;" | 928.30
| 928.30
| style="text-align:center;" | 12/7
| [[12/7]]
| style="text-align:center;" | ^M6
| ^M6
| style="text-align:center;" | upmajor 6th
| upmajor 6th
| style="text-align:center;" | ^B
| ^B
| |  
|  
|-
|-
| style="text-align:center;" | 42
| 42
| style="text-align:center;" | law
| law
| style="text-align:center;" | 950.94
| 950.94
| style="text-align:center;" | 26/15, 125/72
| [[26/15]], [[125/72]]
| style="text-align:center;" | ^^M6
| ^^M6, <br>vvm7
 
| double-up major 6th, <br>double-down minor 7th
vvm7
| ^^B, <br>vvC
| style="text-align:center;" | double-up major 6th,
|  
 
double-down minor 7th
| style="text-align:center;" | ^^B,
 
vvC
| |  
|-
|-
| style="text-align:center;" | 43
| 43
| style="text-align:center;" | ta
| ta
| style="text-align:center;" | 973.58
| 973.58
| style="text-align:center;" | 7/4
| [[7/4]]
| style="text-align:center;" | vm7
| vm7
| style="text-align:center;" | downminor 7th
| downminor 7th
| style="text-align:center;" | vC
| vC
| |  
|  
|-
|-
| style="text-align:center;" | 44
| 44
| style="text-align:center;" | teh
| teh
| style="text-align:center;" | 996.23
| 996.23
| style="text-align:center;" | 16/9
| [[16/9]]
| style="text-align:center;" | m7
| m7
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | C
| C
| |  
|  
|-
|-
| style="text-align:center;" | 45
| 45
| style="text-align:center;" | te
| te
| style="text-align:center;" | 1018.87
| 1018.87
| style="text-align:center;" | 9/5
| [[9/5]]
| style="text-align:center;" | ^m7
| ^m7
| style="text-align:center;" | upminor 7th
| upminor 7th
| style="text-align:center;" | ^C
| ^C
| |  
|  
|-
|-
| style="text-align:center;" | 46
| 46
| style="text-align:center;" | tu
| tu
| style="text-align:center;" | 1041.51
| 1041.51
| style="text-align:center;" | 11/6, 20/11, 729/400
| [[11/6]], [[20/11]], [[729/400]]
| style="text-align:center;" | v~7
| v~7
| style="text-align:center;" | downmid 7th
| downmid 7th
| style="text-align:center;" | ^^C
| ^^C
| |  
|  
|-
|-
| style="text-align:center;" | 47
| 47
| style="text-align:center;" | tuh
| tuh
| style="text-align:center;" | 1064.15
| 1064.15
| style="text-align:center;" | 13/7, 24/13, 50/27
| [[13/7]], [[24/13]], [[50/27]]
| style="text-align:center;" | ^~7
| ^~7
| style="text-align:center;" | upmid 7th
| upmid 7th
| style="text-align:center;" | vvC#
| vvC#
| |  
|  
|-
|-
| style="text-align:center;" | 48
| 48
| style="text-align:center;" | ti
| ti
| style="text-align:center;" | 1086.79
| 1086.79
| style="text-align:center;" | 15/8
| [[15/8]]
| style="text-align:center;" | vM7
| vM7
| style="text-align:center;" | downmajor 7th
| downmajor 7th
| style="text-align:center;" | vC#
| vC#
| |  
|  
|-
|-
| style="text-align:center;" | 49
| 49
| style="text-align:center;" | tih
| tih
| style="text-align:center;" | 1109.43
| 1109.43
| style="text-align:center;" | 40/21, 243/128
| [[40/21]], [[243/128]]
| style="text-align:center;" | M7
| M7
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | C#
| C#
| |  
|  
|-
|-
| style="text-align:center;" | 50
| 50
| style="text-align:center;" | to
| to
| style="text-align:center;" | 1132.08
| 1132.08
| style="text-align:center;" | 48/25, 27/14
| [[48/25]], [[27/14]]
| style="text-align:center;" | ^M7
| ^M7
| style="text-align:center;" | upmajor 7th
| upmajor 7th
| style="text-align:center;" | ^C#
| ^C#
| |  
|  
|-
|-
| style="text-align:center;" | 51
| 51
| style="text-align:center;" | taw
| taw
| style="text-align:center;" | 1154.72
| 1154.72
| style="text-align:center;" | 125/64
| [[125/64]]
| style="text-align:center;" | ^^M7,
| ^^M7, <br>vv8
 
| double-up major 7th, <br>double-down 8ve
vv8
| ^^C#, <br>vvD
| style="text-align:center;" | double-up major 7th,
|  
 
double-down 8ve
| style="text-align:center;" | ^^C#,
 
vvD
| |  
|-
|-
| style="text-align:center;" | 52
| 52
| style="text-align:center;" | da
| da
| style="text-align:center;" | 1177.36
| 1177.36
| style="text-align:center;" | 160/81
| [[160/81]]
| style="text-align:center;" | v8
| v8
| style="text-align:center;" | down 8ve
| down 8ve
| style="text-align:center;" | vD
| vD
| |  
|  
|-
|-
| style="text-align:center;" | 53
| 53
| style="text-align:center;" | do
| do
| style="text-align:center;" | 1200
| 1200.00
| style="text-align:center;" | 2/1
| [[2/1]]
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | perfect 8ve
| perfect 8ve
| style="text-align:center;" | D
| D
| |  
|  
|}
|}
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | quality
! quality
! | [[Kite's color notation|color]]
! [[Kite's color notation|color]]
! | monzo format
! monzo format
! | examples
! examples
|-
|-
| style="text-align:center;" | downminor
| downminor
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | (a, b, 0, 1)
| (a, b, 0, 1)
| style="text-align:center;" | 7/6, 7/4
| 7/6, 7/4
|-
|-
| style="text-align:center;" | minor
| minor
| style="text-align:center;" | fourthward wa
| fourthward wa
| style="text-align:center;" | (a, b) with b &lt; -1
| (a, b) with b &lt; -1
| style="text-align:center;" | 32/27, 16/9
| 32/27, 16/9
|-
|-
| style="text-align:center;" | upminor
| upminor
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | (a, b, -1)
| (a, b, -1)
| style="text-align:center;" | 6/5, 9/5
| 6/5, 9/5
|-
|-
| style="text-align:center;" | downmid
| downmid
| style="text-align:center;" | ilo
| ilo
| style="text-align:center;" | (a, b, 0, 0, 1)
| (a, b, 0, 0, 1)
| style="text-align:center;" | 11/9, 11/6
| 11/9, 11/6
|-
|-
| style="text-align:center;" | upmid
| upmid
| style="text-align:center;" | lu
| lu
| style="text-align:center;" | (a, b, 0, 0, -1)
| (a, b, 0, 0, -1)
| style="text-align:center;" | 12/11, 18/11
| 12/11, 18/11
|-
|-
| style="text-align:center;" | downmajor
| downmajor
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | (a, b, 1)
| (a, b, 1)
| style="text-align:center;" | 5/4, 5/3
| 5/4, 5/3
|-
|-
| style="text-align:center;" | major
| major
| style="text-align:center;" | fifthward wa
| fifthward wa
| style="text-align:center;" | (a, b) with b &gt; 1
| (a, b) with b &gt; 1
| style="text-align:center;" | 9/8, 27/16
| 9/8, 27/16
|-
|-
| style="text-align:center;" | upmajor
| upmajor
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | (a, b, 0, -1)
| (a, b, 0, -1)
| style="text-align:center;" | 9/7, 12/7
| 9/7, 12/7
|}
|}
All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.
All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.


Here are the zo, gu, ilo, lu, yo and ru triads:
Here are the zo, gu, ilo, lu, yo and ru triads:
{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | [[Kite's color notation|color of the 3rd]]
! [[Kite's color notation|color of the 3rd]]
! | JI chord
! JI chord
! | notes as edosteps
! notes as edosteps
! | notes of C chord
! notes of C chord
! | written name
! written name
! | spoken name
! spoken name
|-
|-
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | 6:7:9
| 6:7:9
| style="text-align:center;" | 0-12-31
| 0-12-31
| style="text-align:center;" | C vEb G
| C vEb G
| style="text-align:center;" | Cvm
| Cvm
| style="text-align:center;" | C downminor
| C downminor
|-
|-
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | 10:12:15
| 10:12:15
| style="text-align:center;" | 0-14-31
| 0-14-31
| style="text-align:center;" | C ^Eb G
| C ^Eb G
| style="text-align:center;" | C^m
| C^m
| style="text-align:center;" | C upminor
| C upminor
|-
|-
| style="text-align:center;" | ilo
| ilo
| style="text-align:center;" | 18:22:27
| 18:22:27
| style="text-align:center;" | 0-15-31
| 0-15-31
| style="text-align:center;" | C ^^Eb G
| C ^^Eb G
| style="text-align:center;" | Cv~
| Cv~
| style="text-align:center;" | C downmid
| C downmid
|-
|-
| style="text-align:center;" | lu
| lu
| style="text-align:center;" | 22:27:33
| 22:27:33
| style="text-align:center;" | 0-16-31
| 0-16-31
| style="text-align:center;" | C vvE G
| C vvE G
| style="text-align:center;" | C^~
| C^~
| style="text-align:center;" | C upmid
| C upmid
|-
|-
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | 4:5:6
| 4:5:6
| style="text-align:center;" | 0-17-31
| 0-17-31
| style="text-align:center;" | C vE G
| C vE G
| style="text-align:center;" | Cv
| Cv
| style="text-align:center;" | C downmajor or C down
| C downmajor or C down
|-
|-
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | 14:18:21
| 14:18:21
| style="text-align:center;" | 0-19-31
| 0-19-31
| style="text-align:center;" | C ^E G
| C ^E G
| style="text-align:center;" | C^
| C^
| style="text-align:center;" | C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].


= Relationship to 12-edo =
== Relationship to 12-edo ==
Whereas 12-edo has a circle of twelve 5ths, 53-edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53-edo in a 12-edo-friendly format. Excellent for introducing 53-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
Whereas 12-edo has a circle of twelve 5ths, 53-edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53-edo in a 12-edo-friendly format. Excellent for introducing 53-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
[[File:53-edo spiral.png|702x702px]]
[[File:53-edo spiral.png|702x702px]]


=Just Approximation=
== Just Approximation ==
53edo provides excellent approximations for the classic 5-limit [[just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | interval
! interval
! | ratio
! ratio
! | size
! size
! | difference
! difference
|-
|-
| | perfect fifth
| perfect fifth
| | 3/2
| 3/2
| style="text-align:center;" | 31
| 31
| | −0.07 cents
| −0.07 cents
|-
|-
| | major third
| major third
| | 5/4
| 5/4
| style="text-align:center;" | 17
| 17
| | −1.40 cents
| −1.40 cents
|-
|-
| | minor third
| minor third
| | 6/5
| 6/5
| style="text-align:center;" | 14
| 14
| | +1.34 cents
| +1.34 cents
|-
|-
| | major tone
| major tone
| | 9/8
| 9/8
| style="text-align:center;" | 9
| 9
| | −0.14 cents
| −0.14 cents
|-
|-
| | minor tone
| minor tone
| | 10/9
| 10/9
| style="text-align:center;" | 8
| 8
| | −1.27 cents
| −1.27 cents
|-
|-
| | diat. semitone
| diat. semitone
| | 16/15
| 16/15
| style="text-align:center;" | 5
| 5
| | +1.48 cents
| +1.48 cents
|}
|}


Line 724: Line 665:
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal_kleisma|septimal kleisma]], 225/224.
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal_kleisma|septimal kleisma]], 225/224.


==Selected just intervals by error==
=== Selected just intervals by error ===
{| class="wikitable"
{| class="wikitable center-all"
! colspan="2" |
! colspan="2" |
!prime 2
!prime 2
Line 772: Line 713:
|}
|}
The following table shows how [[15-odd-limit|some prominent just intervals]] are represented in 53edo (ordered by absolute error).  Octave-reduced prime harmonics are '''bolded'''.
The following table shows how [[15-odd-limit|some prominent just intervals]] are represented in 53edo (ordered by absolute error).  Octave-reduced prime harmonics are '''bolded'''.
{| class="wikitable"
{| class="wikitable center-all"
|-
|-
| | '''Interval, complement'''
! '''Interval, complement'''
| | '''Error (abs., in [[cent]]s)'''  
! '''Error (abs., in [[cent]]s)'''  
|-  
|-  
| style="text-align:center;" | '''[[4/3]], [[3/2|3/2]]'''  
| '''[[4/3]], [[3/2|3/2]]'''  
| style="text-align:center;" | '''0.068'''  
| '''0.068'''  
|-  
|-  
| style="text-align:center;" | [[9/8]], [[16/9]]  
| [[9/8]], [[16/9]]  
| style="text-align:center;" | 0.136  
| 0.136  
|-  
|-  
| style="text-align:center;" | [[10/9]], [[9/5]]  
| [[10/9]], [[9/5]]  
| style="text-align:center;" | 1.272  
| 1.272  
|-  
|-  
| style="text-align:center;" | [[15/13]], [[26/15]]  
| [[15/13]], [[26/15]]  
| style="text-align:center;" | 1.316  
| 1.316  
|-  
|-  
| style="text-align:center;" | [[6/5]], [[5/3]]  
| [[6/5]], [[5/3]]  
| style="text-align:center;" | 1.340
| 1.340
|-  
|-  
| style="text-align:center;" | [[13/10]], [[20/13]]  
| [[13/10]], [[20/13]]  
| style="text-align:center;" | 1.384  
| 1.384  
|-  
|-  
| style="text-align:center;" | '''[[5/4]], [[8/5]]'''  
| '''[[5/4]], [[8/5]]'''  
| style="text-align:center;" | '''1.408'''  
| '''1.408'''  
|-  
|-  
| style="text-align:center;" | [[16/15]], [[15/8]]  
| [[16/15]], [[15/8]]  
| style="text-align:center;" | 1.476  
| 1.476  
|-  
|-  
| style="text-align:center;" | [[18/13]], [[13/9]]  
| [[18/13]], [[13/9]]  
| style="text-align:center;" | 2.655  
| 2.655  
|-  
|-  
| style="text-align:center;" | [[13/12]], [[24/13]]  
| [[13/12]], [[24/13]]  
| style="text-align:center;" | 2.724  
| 2.724  
|-  
|-  
| style="text-align:center;" | '''[[16/13]], [[13/8|13/8]]'''  
| '''[[16/13]], [[13/8|13/8]]'''  
| style="text-align:center;" | '''2.792'''  
| '''2.792'''  
|-  
|-  
| style="text-align:center;" | '''[[8/7]], [[7/4]]'''  
| '''[[8/7]], [[7/4]]'''  
| style="text-align:center;" | '''4.759'''  
| '''4.759'''  
|-  
|-  
| style="text-align:center;" | [[7/6]], [[12/7]]  
| [[7/6]], [[12/7]]  
| style="text-align:center;" | 4.827  
| 4.827  
|-  
|-  
| style="text-align:center;" | [[9/7]], [[14/9]]  
| [[9/7]], [[14/9]]  
| style="text-align:center;" | 4.895  
| 4.895  
|-  
|-  
| style="text-align:center;" | [[13/11]], [[22/13]]  
| [[13/11]], [[22/13]]  
| style="text-align:center;" | 5.130
| 5.130
|-  
|-  
| style="text-align:center;" | [[7/5]], [[10/7]]  
| [[7/5]], [[10/7]]  
| style="text-align:center;" | 6.167  
| 6.167  
|-  
|-  
| style="text-align:center;" | [[15/14]], [[28/15]]  
| [[15/14]], [[28/15]]  
| style="text-align:center;" | 6.235  
| 6.235  
|-  
|-  
| style="text-align:center;" | [[15/11]], [[22/15]]  
| [[15/11]], [[22/15]]  
| style="text-align:center;" | 6.445  
| 6.445  
|-  
|-  
| style="text-align:center;" | [[11/10]], [[20/11]]  
| [[11/10]], [[20/11]]  
| style="text-align:center;" | 6.514  
| 6.514  
|-  
|-  
| style="text-align:center;" | [[14/13]], [[13/7]]  
| [[14/13]], [[13/7]]  
| style="text-align:center;" | 7.551  
| 7.551  
|-  
|-  
| style="text-align:center;" | [[11/9]], [[18/11]]  
| [[11/9]], [[18/11]]  
| style="text-align:center;" | 7.785  
| 7.785  
|-  
|-  
| style="text-align:center;" | [[12/11]], [[11/6]]  
| [[12/11]], [[11/6]]  
| style="text-align:center;" | 7.854  
| 7.854  
|-  
|-  
| style="text-align:center;" | '''[[11/8]], [[16/11]]'''  
| '''[[11/8]], [[16/11]]'''  
| style="text-align:center;" | '''7.922'''  
| '''7.922'''  
|-  
|-  
| style="text-align:center;" | [[14/11]], [[11/7]]  
| [[14/11]], [[11/7]]  
| style="text-align:center;" | ''9.961''  
| ''9.961''  
|}
|}


=Linear temperaments=
== Linear temperaments ==
[[List_of_edo-distinct_53et_rank_two_temperaments|List of edo-distinct 53et rank two temperaments]]
[[List_of_edo-distinct_53et_rank_two_temperaments|List of edo-distinct 53et rank two temperaments]]


=Music=
== Music ==
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo_Khramov|Mykhaylo Khramov]]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo_Khramov|Mykhaylo Khramov]]


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